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- Theory Of Computation - TC
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**Sreenivasa Institute of Technology and Management Studies -**- Computer Science Engineering
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SITAMS – B.Tech – III Year - II Sem CSE 13CSE 321 – Theory of Computation Dr. D. Jagadeesan, B.E., M.Tech., Ph.D., Professor in CSE, Unit – I Syllabus : Unit – I : FINITE AUTOMATA Introduction- Strings - Alphabets and languages - Finite state automata – Basic Definitions Deterministic finite automata – Non deterministic finite automata - Equivalence of DFA’s and NFA’s - Finite automata with output – More machines and mealy machines - Minimization of FA Equivalence between two FA’s - NFA with ε –moves - Equivalence of NFA’s with and without ε – Moves. Introduction Definition of TOC TOC describes the basic ideas and models underlying computing. TOC suggests various abstract models of computation, represented mathematically. History of Theory of Computation 1936 Alan Turing invented the Turing machine, and proved that there exists an unsolvable problem. 1940’s Stored-program computers were built. 1943 McCulloch and Pitts invented finite automata. 1956 Kleene invented regular expressions and proved the equivalence of regular expression and finite automata 1956 Chomsky defined Chomsky hierarchy, which organized languages recognized by different automata into hierarchical classes. 1959 Rabin and Scott introduced nondeterministic finite automata and proved its equivalence to (deterministic) finite automata. 1950’s-1960’s More works on languages, grammars, and compilers 1965 Hartmantis and Stearns defined time complexity, and Lewis, Hartmantis and Stearns defined space complexity. 1971 Cook showed the first NP-complete problem, the satisfiability problem. 1972 Karp Showed many other NP-complete problems. 1976 Diffie and Helllman defined Modern Cryptography based on NPcomplete problems. 1978 Rivest, Shamir and Adelman proposed a public-key encryption scheme, RSA. Finite State systems A finite automaton can also be thought of as the device shown below consisting of a tape and a control circuit which satisfy the following conditions: The tape has the left end and extends to the right without an end. The tape is dividing into squares in each of which a symbol can be written prior to the start of the operation of the automaton. The tape has a read only head. The head is always at the leftmost square at the beginning of the operation. The head moves to the right one square every time it reads a symbol. It never moves to the left. When it sees no symbol, it stops and the automaton terminates its operation. 1 / 28

SITAMS – B.Tech – III Year - II Sem CSE 13CSE 321 – Theory of Computation Dr. D. Jagadeesan, B.E., M.Tech., Ph.D., Professor in CSE, Unit – I There is a finite control which determines the state of the automaton and also controls the movement of the head. Basic Definitions Symbol : Symbol is a character. Example : a,b,c,… , 0,1,2,3,….9 and special characters. Alphabet : An alphabet is a finite, nonempty set of symbol. It is denoted by ∑. Example : a) ∑ = {0,1}, the set of binary alphabet. b) ∑ = {a,b……..z}, the set of all lowercase letters. c) ∑ = {+, &,…..}, the set of all special characters. String or Word : A string is a finite set sequence of symbols chosen from some alphabets. Example : a) 0111010 is a string from the binary alphabet ∑ = {0,1} b) aabbaacab is a string from the alphabet ∑ = {a,b,c} Empty String : The empty string is the string with zero occurrences of symbols (no symbols). It is denoted by є. Length of String : The length of a string is number of symbols in the string. It denoted by |w|. Example : w = 010110101 from binary alphabet ∑ = {0,1} Length of a string |w| = 9 2 / 28

SITAMS – B.Tech – III Year - II Sem CSE 13CSE 321 – Theory of Computation Dr. D. Jagadeesan, B.E., M.Tech., Ph.D., Professor in CSE, Unit – I Power of an Alphabet: If ∑ is an alphabet, we can express the set of all strings of certain length from that alphabet by using an exponential notation. It is denoted by ∑k is the set of strings of length k, each of whose symbols is in ∑. Example : ∑ = {0,1} has 2 symbols i) ∑1 = {0,1} ( 21 = 2) ii) ∑2 = {00, 01, 10, 11} ( 22 = 4) iii) ∑3 = {000,001,010,011,100,101,110,111} ( 23 = 8) The set of strings over an alphabet ∑ is usually denoted by ∑*. For instance, ∑* = {0,1}* = {є,0,1,00,01,10,11} (∑*=∑0∑1∑2……) - with є symbol. The set of strings over an alphabet ∑ excluding є is usually denoted by ∑+. For instance, ∑+ = {0,1}+ = {0,1,00,01,10,11} (∑+=∑*- {є} or ∑1∑2∑3…..…) - without є symbol. Concatenation of String Join the two or more strings. Let x and y be two strings. Concatenation of strings x and y is appending symbols of y to right end of x. x = a1a2a3……………an and y = b1b2b3……………bn Concatenation of String xy = a1a2a3……an b1b2b3….…bn Example : s = ababa and t = cdcddc Concatenation st = ababacdcddc Languages: If Σ is an alphabet, and L Σ* then L is a language. Examples: o The set of legal English words o The set of legal C programs o The set of strings consisting of n 0's followed by n 1's { ϵ, 01,0011,000111, …} Operations on Languages Complementation Let L be a language over an alphabet Σ. The complementation of L, denoted byL, is Σ*–L. Union Let L1 and L2 be languages over an alphabet Σ. The union of L1 and L2, denoted by L1L2, is {x | x is in L1 or L2}. Intersection Let L1 and L2 be languages over an alphabet Σ. The intersection of L1 and L2, denoted by L1L2, is { x | x is in L1 and L2}. 3 / 28

SITAMS – B.Tech – III Year - II Sem CSE 13CSE 321 – Theory of Computation Dr. D. Jagadeesan, B.E., M.Tech., Ph.D., Professor in CSE, Unit – I Concatenation Let L1 and L2 be languages over an alphabet Σ. The concatenation of L1 and L2, denoted by L1L2, is {w1w2| w1 is in L1 and w2 is in L2}. Reversal Let L be a language over an alphabet Σ. The reversal of L, denoted by Lr, is {wr| w is in L}. Kleene’s closure Let L be a language over an alphabet Σ. The Kleene’s closure of L, denoted by L*, is {x | for an integer n 0 x = x1 x2 … xn and x1, x2 , …, xn are in L}. ∞ * L = U Li (e.g. a* ={,a,aa,aaa,……}) i=0 Positive Closure Let L be a language over an alphabet Σ. The closure of L, denoted by L+, is { x |for an integer n 1, x = x1x2…xn and x1, x2 , …, xn are in L} ∞ + L = U Li (e.g. a* ={a,aa,aaa,……}) i=1 Finite Automaton Automaton is an abstract computing device. It is a mathematical model of a system, with discrete inputs, outputs, states and set of transitions from state to state that occurs on input symbols from alphabet Σ. It representations: o Graphical (Transition Diagram or Transition Table) o Tabular (Transition Table) o Mathematical (Transition Function or Mapping Function) Formal Definition of Finite Automata A finite automaton is a 5-tuples; they are M=(Q, Σ, δ, q0, F) where Q is a finite set called the states Σ is a finite set called the alphabet δ : Q ×Σ → Q is the transition function q0 ∈ Q is the start state also called initial state F ⊆ Q is the set of accept states, also called the final states Transition Diagram (Transition graph) It is a directed graph associated with the vertices of the graph corresponds to the states of the finite automata. (or) It is a 5-tuple graph used state and edges represent the transitions from one state to other state. 1 Example: 0 1 1 q0 Start or Initial State 0 q1 qq22 Final or Accepting State 4 / 28

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